I think maybe we see this sort of practical if unspoken alliance less often in the United States and Canada than in the UK, but it’s incredibly disturbing regardless of geography.
As far as lies go, one need to go no farther then the homepage of the UK-based, gender-critical LGB Alliance to spot egregious examples.
The scaremongering they engage in to oppose transgender equality is utterly dishonest. For example, the idea that a corporate HR department would fire a lesbian for refusing to date a trans woman in the company is completely ridiculous. Corporations do not try to strong-arm employees into dating one another, and in fact are much more likely to forbid office romances in toto. Yet the LGB Alliance present this hypothetical horror story as a pervasive problem happening right now.
Their claim is a knowing lie.
Their alliance with rightwing conservative religious organizations is well documented. They even work with organizations that have it on their agenda to criminalize or increase criminal penalties for homosexuality in certain parts of the world.
A pointless and dishonest alliance, indeed.